One of the early important results in the study of quantum information is the no-cloning theorem, which tells us that there is no quantum operation that allows us to create multiple copies of an arbitrary quantum state.

This property is very different from what we expect from classical information, which you may reproduce as many times as you wish. For example, you can send a PDF file by email to many recipients while keeping a copy to yourself. The important point is that whatever the contents may be, you can make a duplicate of it.

Now consider a cloning machine

*M*for qubits that can produce identical copies of the states*|u)*and*|d)*:*M |u) |0) = M |u) |u),*

*M |d) |0) = M |d) |d) ,*

where

*|0)*denotes any fixed initial state for*M*. This is necessary in pretty much the same way you would need a blank piece of paper before you can photocopy a printed document.